A376140 The number of divisors of n whose prime factorization has maximum exponent that is smaller than the maximum exponent in the prime factorization of n.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 1, 1, 1, 6, 2, 1, 3, 4, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 4, 4, 1, 1, 8, 2, 4, 1, 4, 1, 6, 1, 6, 1, 1, 1, 8, 1, 1, 4, 6, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 4, 4, 1, 1, 1, 8, 4, 1, 1, 8, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Sep 11 2024

Keywords

• nonn
• easy

Comments

The maximum exponent in the prime factorization of 1 is considered to be A051903(1) = 0.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000005, A005117, A051903, A005361, A362611, A375931, A375932.

Programs

Mathematica

a[n_] := Module[{e = FactorInteger[n][[;;,2]], m}, m = Max[e]; Times@@ ((Min[#, m-1] & /@ e) + 1)]; a[1] = 0; Array[a, 100]

PARI

a(n) = if(n == 1, 0, my(e = factor(n)[,2], m = vecmax(e)); vecprod(apply(x -> 1 + min(x, m-1), e)));

Formula

a(n) = Sum_{d|n} [m(d) = m(n)], where m(n) = A051903(n) and [] is the Iverson bracket.

If n = Product_{i} p_i^e_i (where p_i are distinct primes), then a(n) = Product_{i} (e_i + [e_i < Max_{i}(e_i)]).

a(n) <= 1 if and only if n is squarefree (A005117), and a(n) = 0 only for n = 1.

a(n) < A000005(n).

a(n) = A000005(A375932(n)) * A005361(A375931(n)) for n >= 2.

a(n) = A000005(n) * (A051903(n)/(A051903(n)+1))^A362611(n) for n >= 2.